To start off, we must first assume that morality on some level exists. This is important. If we don't, then we don't have an argument.
So we shall define three terms:
Morally Obligatory - It is morally necessary to do something.
Morally Permissible - It is morally neutral. (A good example is like the act of eating Cheerios)
Morally Forbidden - It is morally wrong to do something. A certain act is "immoral".
Next, you guys will need to know a term that is necessary to this. It is called moral relativity. This entails that a certain act that in one context may be permissible, while in another, is immoral. To bring back the Cheerios reference, it would be morally permissible to refuse to eat Cheerios, and instead eat Frosted Flakes. This choice really has no moral weight on it. However, if a gunman had a gun to your daughter and said that you must eat the Cheerios or he will pull the trigger, refusing to do so is clearly immoral.
The final term you need to know is luxury. A luxury is something that you buy that is morally permissible. This often means that there is nothing morally obligatory or forbidden when you buy that object. So I am driving along, and I see a Dairy Queen. I am just feelin a need for ice cream, and when I buy it, it is indeed delicious. This purchase has no moral relevance, since nothing would have happened if I did not buy it.
Ok, now that we got past that, we can start. Thought-experiment.
Jim is a wall-street broker, he is quite wealthy. He in-fact, is wealthy enough that he has a boat, and he goes down to the Cayman Islands every year to go scuba diving. So he goes down to the islands and he has quite a lot of fun splurging the money he saved for this trip. He parties, he drinks, he sings karaoke. He wakes up that morning and heads out to go scuba diving. There is a underwater cable laying company "Kamo" that is working in the area, and they have O2 kiosks in the area to both provide for their workers, and for selling O2 to people. The kiosks have a bring green button that you press after you have swiped your credit card to fill up your tanks. So, Jim brings over his tanks. He has six of them. He really only needs four at the very most, but he likes bringing along two extra just so he can feel awesome about having extra oxygen tanks. So he swipes his card, then fills up a tank. Then he does it again for the second, third, and fourth tank. Suddenly people are running in a mad panic past him, and he can see the congregating in a large group. He too, follows the crowd, and then he finds out what they are all looking at. There is a live feed depicting that there are multiple submarines just miles off the coast filled with small school children. For whatever reason, the submarines suddenly broke down, and now the children along with the chaperons are trapped under water... and they are running out of air. It is reported that response teams are on their way, but it is likely they will not make it in time. The underwater cable lying company Kamo has announced that it is going to use it's oxygen tubes to transfer oxygen from the kiosks to the submarines. However, the lines do not work very well, only 30% of the oxygen is making it through, and they need everyone's help. If a person swipes their card on the kiosk and presses the yellow button, they can donate their O2 to the submarine to save the children. Jim of course, wishes the children to be saved, as he believes any reasonable person does. So he walks over to his kiosk, and swipes his card he realizes that in his splurging, he only has enough money left on his card for one more transaction. Jim notices his two extra water tanks on the ground, he then thinks of the suffocating children, then presses the green button, filling his O2 tanks. Jim goes to his boat, sets off, and spends many hours scuba diving. He regrets the decision a little bit at first, but his worries are soon washed away by the waves, and in fact he never really thinks about it again. Because of his inaction, three children suffocated.
Now that you hopefully enjoyed that, here is the argument. It is quite straight forward.
Premise one: Jim's actions were immoral. (Obvious Truth)
Premise two: There is no morally relevant difference between Jim’s immoral behavior and my indulging in what I reasonably believe is a luxury instead of trying to prevent the horrific deaths of small innocent children.
Conclusion: Therefore, my indulging in what I reasonably believe is a luxury instead of preventing (or trying to prevent) the deaths of small innocent children is immoral.
The working definition of a morally relevant difference is a difference between my situation and Kent’s situation that clearly illustrates how on earth it could be that my indulging in luxuries is morally permissible, on the one hand, while Jim’s behavior is immoral, on the other hand.
As you can see, this is a tough pill to swallow. If we accept this, it means that we are being immoral several times on a weekly basis. Now, the only way to reject an argument, is to show one of the premises to be false, or to prove an invalid inference. However, to reject the first premise requires an entire argument with premises and inferences and it damn well better be bullet proof, so I suggest we not attempt this. We can't show an inference to be false since the only inference here is Modus ponens, which is always valid. This leaves us with us showing that there is a morally relative difference between the two situations.
Please note that my professor is not going to be unreasonable. He admits that he too, does not like the conclusion. However, only damaging arguments will be suffice, so nitpicking the details of the thought experiment won't work if all that needs to be done is to tweek the thought experiment to render your objection useless.
I honestly have got nothing as of yet, hopefully some of your shall do better.